- Venn diagrams were introduced in 1880 by John Venn (1834-1923).
- These diagrams consist of rectangles and closed curves usually circles.
- The universal set is represented usually by a rectangle and its subsets by circles.
- Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set. For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while another circle may represent the set of all tables

\(A \cup B = \left\{ {x:x \in A{\rm{\; or \;}}x \in B} \right\}\)

- Commutative law : \(X \cup Y = Y \cup X\)
- Associative law : \(\left( {X \cup Y} \right) \cup Z = X \cup \left( {Y \cup Z} \right)\)
- Law of identity element, \(\phi \) is the identity of \( \cup \) : \(X \cup \phi = X\)
- Idempotent law : \(X \cup X = X\)
- Law of U : \(U \cup X = U\)

\(A \cap B = \left\{ {x:x \in A{\rm{\; and\; }}x \in B} \right\}\)

- Commutative law : \(X \cap Y = Y \cap X\)
- Associative law : \(\left( {X \cap Y} \right) \cap Z = X \cap \left( {Y \cap Z} \right)\)
- Law of \( \cap \) and \(U\) : \(\phi \cap X = \phi \) , \(U \cap X = X\)
- Idempotent law : \(X \cap X = X\)
- Distributive law : \(X \cap \left( {Y \cup Z} \right) = \left( {X \cap Y} \right) \cup \left( {X \cap Z} \right)\)

\(A - B = \left\{ {x:x \in A{\rm{ \;and\; }}x \notin B} \right\}\)

- \(A - B \ne B - A\)
- The sets \(\left( {A - B} \right)\) , \(\left( {A \cap B} \right)\) and \(\left( {B - A} \right)\) are mutually disjoint sets.

\(A' = \left\{ {x:x \in U{\rm{\; and\; }}x \notin A} \right\}\) ,obviously \(A' = U - A\)

- Complement laws:
- \(A \cup A' = U\)
- \(A \cap A' = \phi \)

- De Morgan's law:
- \(\left( {A \cup B} \right)' = A' \cap B'\)
- \(\left( {A \cap B} \right)' = A' \cup B'\)

- Law of double complementation: \(\left( {A'} \right)' = A\)
- Laws of empty set and universal set : \(\phi ' = U\) and \(U' = \phi \)

Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find

a) A - B

b) B - A

c) A ∩ B

d) A ∪ B

a) A - B ={1,3,5}

b) B -A ={8}

c) A ∩ B ={2,4,6}

d) A ∪ B={1,2,3,4,5,6,8}

**Notes**- Introduction
- Methods of representing a set
- Types of sets
- Subset
- Subset
- Subset of set of the real numbers
- Interval as subset of R Real Number
- Power Set
- Universal Set
- Venn diagram
- Operation on Sets
- Cardinality of Sets
- Set Theory Symbols

**Assignments**

Class 11 Maths Class 11 Physics Class 11 Chemistry Class 11 Biology

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